Global uniqueness for a two-dimensional inverse boundary value problem

We show that the coefficient gamma(x) of the elliptic equation del . (gamma del u) = 0 in a two-dimensional domain is uniquely determined by the corresponding Dirichlet-to-Neumann map on the boundary, and give a reconstruction procedure. For the equation Sigma partial derivative(i)(gamma(ij)partial derivative(j)u) = 0, two matrix-valued functions gamma(1) and gamma(2) yield the same Dirichlet-to-Neumann map if and only if there is a diffeomorphism of the domain which fixes the boundary and transforms gamma(1) into gamma(2).